Abstract
Thermoelectric effects have been applied to power generators and temperature sensors that convert waste heat into electricity. The effects, however, have been limited to electrons to occur, and inevitably disappear at low temperatures due to electronic entropy quenching. Here, we report thermoelectric generation caused by nuclear spins in a solid: nuclearspin Seebeck effect. The sample is a magnetically ordered material MnCO_{3} having a large nuclear spin (I = 5/2) of ^{55}Mn nuclei and strong hyperfine coupling, with a Pt contact. In the system, we observe lowtemperature thermoelectric signals down to 100 mK due to nuclearspin excitation. Our theoretical calculation in which interfacial Korringa process is taken into consideration quantitatively reproduces the results. The nuclear thermoelectric effect demonstrated here offers a way for exploring thermoelectric science and technologies at ultralow temperatures.
Introduction
Thermoelectric effects enable the direct conversion of thermal energy into electric energy, promising for power generation and waste heat recovery. Most of the prevalent thermoelectric generators have relied on the Seebeck effect, which is the generation of an electric voltage by placing a conductor junction in a temperature gradient^{1,2,3}. Recently, in the study of spintronics, a spin analog of the Seebeck effect—the spin Seebeck effect (SSE)^{4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}—was discovered. The SSE is the generation of a spin current, a flow of spin angular momentum, as a result of a temperature gradient applied across a junction consisting of a magnet and a metal^{17}. In electronic SSE, a thermally generated magnon flow in a magnet injects a conductionelectron spin current into the adjacent metal via the interfacial electronic spin exchange^{8,9,13,16}. The spin current injected into a metal can be converted into a voltage by the inverse spin Hall effect (ISHE)^{21,22,23,24}, enabling unexplored approaches toward thermoelectric conversion and energyharvesting technologies^{10,17,18}.
Up to now, all the thermoelectric effects have been an exclusive feature of electrons^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}. At low temperatures, however, their efficiency is dramatically suppressed, as the thermodynamic entropy of electrons steeply reduces to zero when approaching absolute zero temperature, according to the third law of thermodynamics. In the case of Seebeck effects in semiconductors, the entropy reduction is related to the exponential suppression of the thermally excited charge carriers^{2}, whereas, in SSEs, it is related to the freezing out of spin fluctuations (magnons)^{15,16,17}. Seebeck effects in metals are also suppressed at low temperatures, as the efficiency is governed by \({k}_{\mathrm{B}}T/{\epsilon }_{\mathrm{F}}\)^{1}, where k_{B} is the Boltzmann constant, T the environmental temperature, and \({\epsilon }_{\mathrm{F}}\) the Fermi energy. Therefore, so far, the thermoelectric applications have been limited to higher temperatures, as no mechanism in the ultralow temperature regime (~mK range) has been found.
In solids, there is a hitherto unexplored entropy carrier that is well activated even at ultralow temperatures: a nuclear spin. Because of its tiny gyromagnetic ratio γ_{n} (~10^{3} times less than that of electrons^{25} γ_{e}), a nuclear spin exhibits much lower excitation energy than that of electron spins in ambient fields, allowing its thermal agitation. Here, a question arises: can nuclear spins generate thermoelectric effects? If spin angular momentum can be extracted from nuclei in the form of an electron spin current under a temperature bias, it should generate a thermoelectric voltage via the ISHE in an attached metal, realizing allsolidstate thermoelectricity based on atomic nuclei.
Here we report an observation of the nuclear SSE (Fig. 1a) in a heterostructure composed of a Pt film and a crystal of easyplane canted antiferromagnetic MnCO_{3}^{26,27,28} (Fig. 1d). In MnCO_{3}, ^{55}Mn nuclei, a 100% naturalabundance isotope, carry a large spin I of 5/2 and exhibit strong hyperfine coupling with electrons, which allows spin transfer between nuclei and electrons as recently found in the spin pumping measurements under nuclear magnetic resonance^{28}. In MnCO_{3} single crystals covered with Pt films, we found a strong thermoelectric signal enhancement down to 100 mK (Fig. 1e), as shown below, which demonstrates thermoelectric generation at ultralow temperatures. The experimental results are quantitatively reproduced by a theory for nuclear SSE in which the Korringa process^{29} due to the hyperfine coupling between nuclear spins in the MnCO_{3} and conductionelectron spins in the attached Pt is taken into consideration (Fig. 1a).
Results
Sample and measurement setup
We have used the ISHE^{21,22,23,24} in the Pt film to detect a spin current injected into the film. The ISHE converts a spin current, J_{s}, into an electric field, E_{ISHE}, through the spin–orbit interaction of conduction electrons, which can be strong in heavy metals such as Pt^{10,17}. When a spin current induced by a nuclear SSE carries spin polarization \(\hat{\bf{s}}\) parallel to the net nuclearspin polarization I along the spatial direction J_{s}, E_{ISHE} is given by (Fig. 1b)
where ρ and θ_{SHE} are the resistivity and the spin Hall angle of the Pt layer, respectively. By measuring E_{ISHE}, nuclear SSEs can be detected electrically. We note that, as the spin current J_{s} flows normal to the Pt/MnCO_{3} interface (\({\bf{J}}_{\mathrm{s}}\parallel {\bf{x}}\)), the resultant voltage signal V is maximal for \(\hat{\bf{s}}\,({\parallel} {\bf{I}})\parallel {\bf{z}}\), when E_{ISHE} is measured along the y direction shown in Fig. 1b. However, because of the tiny Zeeman coupling of nuclear spins, it is challenging to control nuclearspin polarization by using B, unlike electronic magnetization in conventional magnets. Nevertheless, we can overcome the difficulty by using a magnetic ordered material carrying a large nuclear spin and strong hyperfine coupling. We have noticed that an antiferromagnet MnCO_{3} (\(I=5/2\))^{26,27,28} satisfies all such conditions. Below the Néel temperature (\({T}_{\mathrm{N}}=35\,{\rm{K}}\)) of MnCO_{3}, the Mn^{2+} sublattice magnetizations M_{1} and M_{2} are aligned in the (111) plane and canted slightly from the collinear antiferromagnetic configuration due to the bulk Dzyaloshinskii–Moriya interaction^{26} (see Fig. 1d and Supplementary Note 1). The hyperfine (Overhauser) fields B_{hf} acting on the ^{55}Mn sublattice nuclear spins I_{1} and I_{2} due to M_{1} and M_{2} reach as large as 57 T^{28}, which induce nuclearspin polarization (~40% at 100 mK) and orient I_{1} and I_{2} along the M_{1} and M_{2} directions, respectively^{30}, as shown in Fig. 1d. Moreover, the net nuclearspin polarization (I_{1} and I_{2}) direction can be controlled by applying B, as the canting angle θ of M_{1} and M_{2} changes with B, owing to the very weak magnetocrystalline anisotropy (~0.1 mT within the easy plane^{26}, see Fig. 1d). The advantage enables us to prepare a controllable nuclearspin polarization in MnCO_{3}, making nuclear SSE experiments feasible.
The SSE devices used in the present study consist of a 10 nmthick Pt strip [200 μm long (l) and 100 nm wide (w)] deposited on the top of an insulating MnCO_{3} (111) (\(3\times 3\times 0{.5\,{\rm{mm}}}^{3}\)) crystal (see “Methods” and Supplementary Note 2). The Pt strip acts as a heater as well as a spinvoltage converter based on the ISHE for measuring nuclear SSEs: by applying an a.c. current I_{c} \((=\sqrt{2}{I}_{{\rm{rms}}}\,\sin \,{{\omega }}t)\,\) to the Pt strip to generate heat and measuring the second harmonic voltage V generated in the Pt by a lockin technique^{11,14}, we can selectively detect the ISHE voltage arising from the temperature drop across the Pt/MnCO_{3} interface induced by the Joule heating ∝ I^{2}_{rms} of the applied current. The SSE experiments were conducted with a ^{4}He cryostat down to 1.82 K using a Pt/MnCO_{3} device named Device 1 and with a ^{3}He–^{4}He dilution refrigerator down to 100 mK using a similar device named Device 2. The Pt/MnCO_{3} devices were mounted in the cryostats and the magnetic field B was applied along the z direction as shown in Fig. 1b. Further details are described in “Methods.”
Observation of nuclear SSE
In Fig. 2a, we show the voltage V data measured at \(T=20\,{\rm{K}}\) and 1.82 K for the Pt/MnCO_{3} Device 1. At 20 K, no voltage signal appears with the application of B. On the other hand, at a lower temperature \(T=1.82\,{\rm{K}}\), an unconventional voltage signal shows up. The sign of V/I^{2}_{rms} reverses by reversing the B direction. The signal intensity increases monotonically with increasing B from zero and it takes a broad peak at around 4 T. For further high B, V/I^{2}_{rms} starts to decrease. We confirmed that the observed signal shares the characteristic feature of ISHE induced by SSE^{10,17}; V appears only when a heat current is applied and the V intensity scales linearly with the heat power ∝ I^{2}_{rms}. The signal intensity is maximal when \({\bf{B}}\parallel {\bf{z}}\) but vanishes when \({\bf{B}}\perp z\), consistent with the prediction of Eq. (1). The sign of V reverses when the Pt strip (\({\theta }_{{\rm{SHE}}} > 0\)) is replaced with tungsten exhibiting a negative^{10} θ_{SHE}. The results confirm that the voltage signal is induced by thermally driven spin currents and ISHE (see Supplementary Notes 3–5 for details).
Surprisingly, the signal intensity persists down to the ultralow temperature regime. Figure 2c, d show the B dependence of V/I^{2}_{rms} at 1.8 K < T < 50 K for Device 1 and at 100 mK < T < 1.6 K for Device 2, respectively. With decreasing temperature T starting from 50 K, the SSE signal appears below ~10 K and its intensity dramatically increases by further decreasing T (see Fig. 2c and b, in which the T dependence of the maximum V/I^{2}_{rms} is plotted). Importantly, the signal intensity continues to increase down to ultralow temperatures on the order of ~100 mK (see Fig. 2d and the inset to Fig. 2b). Moreover, the signal persists in the higher field range up to 14 T even at such ultralow temperatures, which is totally distinct from the conventional SSE driven by electronic magnetization dynamics. For instance, in ferrimagnetic Y_{3}Fe_{5}O_{12}, the SSE intensity decreases monotonically with decreasing temperature below 20 K and completely disappears below 5 K at 14 T due to the freezing out of magnons^{15,16,17}. The maximum output of Device 2 normalized by its electrical resistance \({R}_{{\rm{Pt}}}\), heating power R_{Pt}I^{2}_{rms}, and geometric factor \({l}^{1}\) is as large as V_{max}l/(R^{2}_{Pt}I^{2}_{rms}) ∼ 58 nA mW^{−1} at 101 mK, which is nearly two orders of magnitude higher than that of a prototypical roomtemperature SSE device made of Pt/Y_{3}Fe_{5}O_{12} (~1 nA mW^{−1}) having the same electrode and heater dimensions [see Supplementary Note 7 and Eq. (3) in Supplementary Note 9 for details].
Nuclear and electronspin excitation spectra in MnCO_{3}
We now discuss the results in terms of the nuclear and electronspin excitation features in MnCO_{3}. In Fig. 1c, we show the electronic and nuclearspin excitation spectra in MnCO_{3}^{26,27,28} for several fields at T = 100 mK, whose thermal energy k_{B}T is depicted as the green dashed line. Above k_{B}T, thermal excitation is exponentially suppressed. The lower branch at around 600 MHz, corresponding to 30 mK, originates from the nuclearspin excitation ω_{n}, whose excitation gap is dominated by the strong hyperfine internal field B_{hf} = ω_{n}/γ_{n} ∼ 57 T. The upper branches represent the electronic spinwave modes ω_{mk}, which shift toward higher frequencies with increasing B due to the strong Zeeman effect. At B = 14 T, the electronic spin excitation gap \({\omega }_{{\rm{m0}}}(\approx {\gamma }_{\rm{e}}B)\) is ~19 K, two orders of magnitude greater than the thermal energy = 100 mK, resulting in a negligibly small value of the Boltzmann factor \(\exp (\hslash {\omega }_{{\rm{m0}}}/{k}_{\mathrm{B}}T)\,{ \sim 10}^{82}\ll 1.\) If the SSE we measured were driven by the electronic spinwave modes, the SSE signal would be completely suppressed by applying a strong field of 14 T, as with the conventional SSE of Y_{3}Fe_{5}O_{12}^{15}. This clearly shows the irrelevance of the electronic SSE to the observed signal at low temperatures. On the other hand, the nuclearspin mode can be greatly excited even by such a small thermal energy of ~100 mK and it remains almost unaffected by the applied B due to the tiny Zeeman effects, much weaker than the hyperfine internal field ~ 57 T (Fig. 1c); the nuclear spins can contribute to SSEs even in such a lowT and highB environment. The results also suggest that direct coupling between nuclear spins in the MnCO_{3} and electrons in the Pt at the interface should be responsible for the SSE, rather than the interfacial electronic exchange mediated by the gapped magnons under strong magnetic fields.
Theoretical model for nuclear SSE
We theoretically model the nuclear SSE in which direct nuclearelectron coupling due to the Korringa process^{29} is taken into consideration. In the model, the spin current \({J}_{{\rm{ne}}}\) is generated by the interfacial hyperfine interaction between nuclear spins in the MnCO_{3} and conductionelectron spins in the Pt under the temperature bias \({T}_{\mathrm{e}}{T}_{\mathrm{p}}\) (see Fig. 3a and Supplementary Note 9 for details). Here, T_{e} and T_{p} represent effective temperatures for electrons in the Pt and phonons in the MnCO_{3} near the interface, respectively. The nuclearspin current \({J}_{{\rm{ne}}}\) arises in proportion to the effective temperature difference between the electrons in the Pt (T_{e}) and nuclei in the MnCO_{3} (T_{n}): \({J}_{{\rm{ne}}}={\Gamma }_{{\rm{ne}}}{k}_{\mathrm{B}}({T}_{\mathrm{e}}{T}_{\mathrm{n}})\). Here T_{n} may deviate from the electron T_{e} due to the nuclearphonon thermalization in MnCO_{3} given by \({J}_{{\rm{np}}}={\Gamma }_{\rm{np}}{k}_{\mathrm{B}}({T}_{\mathrm{n}}{T}_{\mathrm{p}}),\,\) resulting in the finite spin current \({J}_{{\rm{ne}}}\). The expression for the nuclear SSE coefficient reads
where \({g}_{{\rm{n}}}^{\uparrow \downarrow }\) is the nuclear spinmixing conductance per unit area, \(\chi\) the normalized antiferromagnetic transverse susceptibility such that \(\theta =\chi b\) is the canting angle, \(b\equiv \hslash {\gamma }_{\mathrm{e}}{s}_{\mathrm{e}}B\) the normalized magnetic field with saturated spin density s_{e} (\({s}_{\mathrm{e}}\equiv S/V\), for volume per site V), and T the average temperature. The bracketed expression in Eq. (2) is evaluated as \(({T}_{\mathrm{e}}{T}_{\mathrm{n}})/({T}_{\mathrm{e}}{T}_{\mathrm{p}})={(1+{\Gamma }_{{\rm{ne}}}/{\Gamma }_{{\rm{np}}})}^{1}\) from the steadystate condition \({J}_{{\rm{ne}}}={J}_{{\rm{np}}}\)^{31}. Here, Γ_{ne} ∝ 1/T [Eq. (1) shown in Supplementary Note 9] and Γ_{np} ∝ 1/Tω^{2}_{m0} is derived by Fermi’s Golden rule for the nuclearphonon thermalization rate mediated by virtual magnons (see Supplementary Note 9), which allow us to evaluate the B dependence of \({T}_{\mathrm{e}}{T}_{\mathrm{n}}\). As shown in Fig. 3b, it is maximal at zero field by the strong thermalization (i.e., \({T}_{\mathrm{n}} \sim {T}_{\mathrm{p}}\)) and decreases gradually with B. There is a crossover field B_{c}, marked by Γ_{np} falling below Γ_{ne} (see the results at \(T=100\,{\rm{mK}}\) and 1 K in Fig. 3b). In Fig. 3c, we compare the B dependence of the experimental V/I^{2}_{rms} (blue plots) for Device 2 and calculated V/I^{2}_{rms} based on the nuclear SSE \({{\cal{S}}}_{{\rm{n}}}\) (red solid curve) at \(T=100\,{\rm{mK}}\). Of important note, the experimental data are quantitatively reproduced by the calculation. Such agreement is confirmed also for other B and T regions (see Fig. 3d, e). A nonmonotonic field response of V now becomes evident: for \(B\ll {B}_{\mathrm{c}}\), the SSE signal increases in proportion to B (\({{\cal{S}}}_{\mathrm{n}}\propto B\)), owing to the increased canting angle, and it takes a maximum at \(B \sim {B}_{\mathrm{c}}\). For \(B\gg {B}_{\mathrm{c}}\), the SSE signal decreases monotonically with B (\({{\cal{S}}}_{\mathrm{n}}\propto {B}^{1}\)) due to the reduction of thermal nonequilibrium \({T}_{\mathrm{e}}{T}_{\mathrm{n}}\) (\(\propto {B}^{2}\)) between the electron and nuclear systems (see Fig. 3b). We also evaluated the electronic SSE, \({{\cal{S}}}_{{\rm{m}}}\), driven by the antiferromagnetic spinwave mode ω_{mk} (see Supplementary Note 9 for details) and found that its intensity, as well as B and T dependencies, do not explain the experimental results (see Fig. 3c and its inset), which confirm that the nuclear SSE dominates the observed SSE.
Discussion
We finally discuss the difference between the previous nuclearspin pumping^{28} and the present nuclear SSE. For the nuclearspin pumping, the measured voltage is maximal at a relatively low field of ~0.3 T and then starts to decrease with B. In such a lowB range, the excitation gap of electronic spinwave branch in MnCO_{3} is comparable to that of the nuclear spins, and a nuclearspin wave, hybridized electronic spinwave and nuclearspin mode^{32,33,34}, is excited. The experimental result in ref. ^{28} was thereby attributed to the coherent nuclear spinwave formation, the electronic (magnetization) component of which pumps a spin current into an adjacent metallic layer in analogy with the conventional electronic spin pumping for a magnet/metal bilayer. On the other hand, the present nuclear SSE increases with B up to around 4–5 T, whereas nuclearelectronic hybridization is quickly suppressed as the electronic spin waves become gapped out. This suggests that a different physical mechanism governs the nuclear SSE, which is reasonable, as the nuclear pumping in the SSE is not limited to a coherent longwavelength dynamics. We thus develop a nuclear SSE theory in terms of interfacial Korringa relaxation, in which nuclearspin fluctuation directly transmits a spin current into an attached metallic layer via interfacial hyperfine interaction, and found quantitative agreement between the experiment and calculation. The Korringa mechanism does not need strong nuclearelectronic spin hybridization in the magnetic layer and also electronic spin transfer at the interface. This may extend a class of materials applicable for nuclear spintronics; materials having magnetic elements with nuclear spins and strong hyperfine interaction, such as ^{55}Mn and ^{59}Co (both of which are 100% natural abundance), can be potential sources of nuclearspin currents.
In summary, we demonstrated the thermoelectric conversion driven by nuclear spin: the nuclear SSE. The nuclear SSE is enhanced at ultralow temperatures, in stark contrast to conventional electronbased thermoelectricity. It is surely worthwhile to explore nuclear SSEs in other systems to show the generality of the phenomenon. Materials of interest include easyaxis antiferromagnetic insulators having a large nuclear spin and exhibiting a spinflop transition, at which the electronic magnon gap comes close to the low energy scales relevant to the nuclear dynamics^{35}; for the nuclear SSE, this is instrumental in thermal equilibration of the nuclei within the magnetic material.
The present work may serve as the bridge between nuclearspin science and thermoelectricity and marks the beginning of a research field “Nuclear thermoelectricity”. It is also worth exploring the reciprocal of the nuclear SSE, as it will be applied to making a nuclear heat pump working at ultralow temperatures.
Methods
Sample preparation
We used singlecrystalline MnCO_{3} slabs with a size of 3 × 3 × 0.5 mm^{3}, which are commercially available from SurfaceNet. The largest plane is (111) in the rhombohedral representation^{36,37}. On the top of the (111) plane of the MnCO_{3} slabs, 10 nmthick Pt strips (200 μm long and nominally 100 nm wide) were patterned by electron beam lithography and liftoff methods (see also Supplementary Note 2). The Pt strips were deposited by magnetron sputtering in a 10^{−1} Pa Ar atmosphere. For a control experiment, we also prepared W/MnCO_{3} devices, where the Pt strips are replaced with 10 nmthick W strips (200 μm long and 500 nm wide) exhibiting a negative θ_{SHE}^{10,38,39}.
SSE measurement
We measured the SSE by a standard lockin technique^{11,14,40,41} with a PPMS (Quantum Design) from 1.8 to 50 K and a ^{3}He–^{4}He dilution refrigerator (KelvinoxMX200, Oxford Instruments; cooling power of 200 µW at 100 mK) from 100 mK to 10 K. An a.c. charge current (\({I}_{\mathrm{c}}=\sqrt{2}{I}_{{\rm{rms}}}\,\sin \,{{\omega }}{t}\)) was applied to the Pt strip with a current source (6221, Keithley) and the generated voltage V across the strip was recorded with a lockin amplifier (LI5640, NF Corporation). For the measurements with the dilution refrigerator, we further introduced a voltage preamplifier (1201, DL Instruments) and a programmable filter (3625, NF Corporation) to reduce signal noise. The typical a.c. charge current property is as follows: the rootmeansquare (rms) amplitude I_{rms} of 0.1–5 μA and the frequency \(\omega /2{\rm{\pi }}\) of 13.423 Hz. All the V–B data are antisymmetrized with respect to the magnetic field B.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes used in theoretical simulations and calculations are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank Y. Chen, J. Lustikova, T. Hioki, N. Yokoi, H. Chudo, M. Imai, K. Sato, and G. E. W. Bauer for fruitful discussions and T. Nojima for his valuable comments on lowtemperature experiments. This work was supported by JST ERATO “Spin Quantum Rectification Project” (JPMJER1402), JST CREST (JPMJCR20C1 and JPMJCR20T2), JSPS KAKENHI (JP19H05600, JP19K21031, JP20H02599, JP20K22476, and JP20K15160), MEXT [Innovative Area “Nano Spin Conversion Science” (JP26103005)], and Daikin Industries, Ltd. The work at UCLA was supported by the US Department of Energy, Office of Basic Energy Sciences under Award number DESC0012190. K.O. acknowledges support from GPSpin at Tohoku University. R.R. acknowledges support from the European Commission through the project 734187SPICOLOST (H2020MSCARISE2016), the European Union’s Horizon 2020 research and innovation program through the Marie SklodowskaCurie Actions grant agreement SPEC number 894006 and the Spanish Ministry of Science (RYC 2019026915I).
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T.K., T.S., and K.O. fabricated the devices. T.K. and T.M. constructed the experimental setup with the help of R.R. and K.O. T.K., T.M., H.I., T.S., K.T., and S.D. performed the experiments and collected the data. T.K. and H.I. analyzed the data with input from D.R. D.R. and Y.T. developed the theoretical explanations. E.S. and Y.T. conceived and supervised the project. T.K. and D.R. wrote the paper with review and input from E.S. and Y.T. T.K., D.R., H.I., T.M., T.S., K.T., S.D., K.O., R.R., S.T., Y.S., Y.T., and E.S. discussed the results and commented on the manuscript.
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Kikkawa, T., Reitz, D., Ito, H. et al. Observation of nuclearspin Seebeck effect. Nat Commun 12, 4356 (2021). https://doi.org/10.1038/s41467021246236
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DOI: https://doi.org/10.1038/s41467021246236
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